The stability threshold for 2D MHD equations around Couette with general viscosity and magnetic resistivity

We address a threshold problem of the Couette flow $(y,0)$ in a uniform magnetic field $(\beta ,0)$ for the 2D MHD equation on $\mathbb{T}\times \mathbb{R}$ with fluid viscosity $\nu$ and magnetic resistivity $\mu$. The nonlinear enhanced dissipation and inviscid damping are also established. In particularly, when $0<\nu \le {\mu }^3\le 1$, we get a threshold ${\nu }^{\frac{1}{2}}{\mu }^{\frac{1}{3}}$ in $H^N(N\ge 4)$. When $0<{\mu }^3\le \nu \le 1$, we obtain a threshold $\min \{{\nu }^{\frac{1}{2}},{\mu }^{\frac{1}{2}}\}\min \{1,{\nu }^{-1}{\mu }^{\frac{1}{3}}\}$, hence improving the results in [19,14,21].